**Introduction.** Let us hark back to the first grade
when the only numbers you knew were the

ordinary everyday integers. You had no trouble solving problems in which you
were, for instance,

asked to find a number x such that 3x = 6. You were quick to answer ”2”. Then,
in the second

grade, Miss Holt asked you to find a number x such that 3x = 8. You were
stumped—there was

no such ”number”! You perhaps explained to Miss Holt that 3(2) = 6 and 3(3) = 9,
and since 8 is

between 6 and 9, you would somehow need a number between 2 and 3, but there
isn’t any such

number. Thus were you introduced to ”fractions.”

These fractions, or rational numbers, were defined by Miss
Holt to be ordered pairs of

integers—thus, for instance, (8, 3) is a rational number. Two rational numbers (n,m)
and (p, q)

were defined to be equal whenever nq = pm. (More precisely, in other words, a
rational number is

an equivalence class of ordered pairs, etc.) Recall that the arithmetic of these
pairs was then

introduced: the sum of (n,m) and (p,q) was defined by

and the product by

Subtraction and division were defined, as usual, simply as the inverses of the two operations

In the second grade, you probably felt at first like you
had thrown away the familiar integers and

were starting over. But no. You noticed that (n, 1) + (p, 1) = (n + p, 1) and
also

(n, 1)(p,1) = (np, 1). Thus the set of all rational numbers whose second
coordinate is one behave

just like the integers. If we simply abbreviate the rational number (n, 1) by n,
there is absolutely no

danger of confusion: 2 + 3 = 5 stands for (2, 1) + (3, 1) = (5, 1). The equation
3x = 8 that started

this all may then be interpreted as shorthand for the equation (3, 1)(u, v) =
(8, 1), and one easily

verifies that x = (u, v) = (8, 3) is a solution. Now, if someone runs at you in
the night and hands

you a note with 5 written on it, you do not know whether this is simply the
integer 5 or whether it is

shorthand for the rational number (5, 1). What we see is that it really doesn’t
matter. What we

have ”really” done is expanded the collection of integers to the collection of
rational numbers. In

other words, we can think of the set of all rational numbers as including the
integers–they are

simply the rationals with second coordinate 1.

One last observation about rational numbers. It is, as
everyone must know, traditional to write the

ordered pair (n,m) as n/m. Thus n stands simply for the rational number n/1,
etc.

Now why have we spent this time on something everyone
learned in the second grade? Because

this is almost a paradigm for what we do in constructing or defining the
so-called complex

numbers. Watch.

Euclid showed us there is no rational solution to the
equation x^2 = 2. We were thus led to defining

even more new numbers, the so-called real numbers, which, of course, include the
rationals. This is

hard, and you likely did not see it done in elementary school, but we shall
assume you know all

about it and move along to the equation x^2 = -1. Now we define **complex
numbers**. These are

simply ordered pairs (x, y) of real numbers, just as the rationals are ordered
pairs of integers. Two

complex numbers are equal only when there are actually the same–that is (x, y) =
(u. v) precisely

when x = u and y = v. We define the sum and product of two complex numbers:

and

As always, subtraction and division are the inverses of these operations

Now let’s consider the arithmetic of the complex numbers with second coordinate 0:

and

Note that what happens is completely analogous to what
happens with rationals with second

coordinate 1. We simply use x as an abbreviation for (x, 0) and there is no
danger of confusion:

x + u is short-hand for (x, 0) + (u, 0) = (x + u, 0) and xu is short-hand for
(x, 0)(u, 0). We see that

our new complex numbers include a copy of the real numbers, just as the rational
numbers include

a copy of the integers.

Next, notice that x(u, v) = (u, v)x = (x, 0)(u, v) = (xu,
xv). Now then, any complex number

z = (x, y) may be written

When we let α = (0, 1), then we have

Now, suppose z = (x, y) = x + αy and w = (u, v) = u + αv. Then we have

We need only see what α^2 is: α^2 = (0, 1)(0, 1) = (-1,
0), and we have agreed that we can safely

abbreviate (-1, 0) as -1. Thus, α^2 = -1, and so

and we have reduced the fairly complicated definition of
complex arithmetic simply to ordinary real

arithmetic together with the fact that α^2 = -1.

Let’s take a look at division–the inverse of
multiplication. Thus z/w stands for that complex number

you must multiply w by in order to get z . An example:

Note this is just fine except when u^2 + v^2 = 0; that is,
when u = v = 0. We may thus divide by any

complex number except 0 = (0, 0).

One final note in all this. Almost everyone in the world
except an electrical engineer uses the letter

i to denote the complex number we have called α. We shall accordingly use i
rather that α to stand

for the number (0, 1).

**Exercises**

1. Find the following complex numbers in the form x + iy:

2. Find all complex z = (x, y) such that

3. Prove that if wz = 0, then w = 0 or z = 0.

**Geometry.** We now have this collection of all
ordered pairs of real numbers, and so there is an

uncontrollable urge to plot them on the usual coordinate axes. We see at once
then there is a

one-to-one correspondence between the complex numbers and the points in the
plane. In the usual

way, we can think of the sum of two complex numbers, the point in the plane
corresponding to

z + w is the diagonal of the parallelogram having z and w as sides:

We shall postpone until the next section the geometric
interpretation of the product of two complex

numbers.

The modulus of a complex number z = x + iy is defined to
be the nonnegative real number

, which is, of course, the length of the
vector interpretation of z. This modulus is

traditionally denoted |z|, and is sometimes called the **length** of z. Note
that and

so is an excellent choice of notation for
the modulus.

The **conjugate**
of a complex number z = x + iy is defined by .
Thus .

Observe that if z = x + iy and w = u + iv, then

In other words, the conjugate of the sum is the sum of the
conjugates. It is also true that .

If z = x + iy, then x is called the real part of z, and y is called the
imaginary part of z. These are

usually denoted Rez and Imz, respectively. Observe then that
Rez and Imz.

Now, for any two complex numbers z and w consider

In other words,

the so-called **triangle inequality**. (This inequality
is an obvious geometric fact–can you guess why

it is called the triangle inequality?)

**Exercises**

4. a)Prove that for any two complex numbers,

b)Prove that

5. Prove that |zw| = |z||w| and that

6. Sketch the set of points satisfying

**Polar coordinates.** Now let’s look at polar
coordinates (r, θ) of complex numbers. Then we may

write z = r(cosθ + i sinθ). In complex analysis, we do not allow r to be
negative; thus r is simply

the modulus of z. The number θ is called an argument of z, and there are, of
course, many

different possibilities for θ. Thus a complex numbers has an infinite number of
arguments, any two

of which differ by an integral multiple of 2π. We usually write θ = arg z. The
**principal argument**

of z is the unique argument that lies on the interval

**Example. **For 1 - i, we have

etc., etc., etc. Each of the numbers
is an argument of 1 - i, but the principal

argument is

We have the nice result that the product of two complex
numbers is the complex number whose

modulus is the product of the moduli of the two factors and an argument is the
sum of arguments

of the factors. A picture:

We now define exp(iθ), or by

We shall see later as the drama of the term unfolds that
this very suggestive notation is an excellent

choice. Now, we have in polar form

where r = |z| and θ is any argument of z. Observe we have just shown that

It follows from this that . Thus

It is easy to see that

**Exercises**

7. Write in polar form

8. Write in rectangular form—no decimal approximations, no trig functions:

9. a) Find a polar form of

b) Use the result of a) to find

10. Find the rectangular form of

11. Find all z such that z^3 = 1. (Again, rectangular form, no trig functions.)

12. Find all z such that z^4 = 16i. (Rectangular form, etc.)

**George Cain**

School of Mathematics

Georgia Institute of Technology